Does my Cayman company have its own CRS or FATCA reporting?

Only if it counts as a financial institution, such as an investment fund. Those entities register with the DITC and report accounts under the Common Reporting Standard and FATCA. A simple holding company is usually not a financial institution, but its bank will still report it, so the owners' home tax offices should expect to see it (confirm your classification with your specialist).

Source: Common Reporting Standard, Department for International Tax Cooperation, Cayman Islands